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                <h1 id="title" titleSize="">
                    Type
                </h1>
            
            <h1 id="motivation--definition">Motivation &amp; Definition</h1>
<p>Let $M$ be a model of a first-order <a href=https://zhaoshenzhai.github.io/mathwiki/theory.md class="internalLink references" title="Theory" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/theory.md&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:05:04-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:05:04-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:05:04-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Theory&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/theory&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-12-15T16:05:04-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:05:04-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:05:04-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Theory&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/theory&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/theory.md&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:05:04-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:05:04-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:05:04-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Theory&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/theory&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">theory</a> $T$. What are the possible first-order properties satisfied elements (or tuples) in $M$?</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>Let $n\geq0$. An <em>$n$-type</em> is a set $p(\bar{x})$ of $L$-formulas such that the $L$-theory $\ex\bar{x}p(\bar{x})\cup\Th M$ is satisfiable.</div>

<p>It will be useful to also consider types with <em>parameters</em> over some subset $A\subseteq M$, which is defined by considering the $L_A$-structure $M$; that is, an <em>$n$-type over $A$</em> is a set $p(\bar{x})$ of $L_A$-formulas such that the $L_A$-theory $\ex\bar{x}p(\bar{x})\cup\Th_AM$ is satisfiable.
<br>
  Every type $\bar{a}\in M^n$ defines the type $\tp^M(\bar{a}/A)\coloneqq\l\{\phi\in L_A\st M\models\phi(\bar{a})\r\}$ over $A$. An $n$-type $p(\bar{x})$ of $M$ over $A$ is:</p>
<ul>
<li><em>complete</em> if either $\phi\in p$ or $\lnot\phi\in p$ for all $L_A$-formulas $\phi$. We write $S^M_n(A)$ for the set of all complete $n$-types over $A$;</li>
<li><em>realized by $\bar{a}\in M^n$</em> if $p\subseteq\tp^M(\bar{a}/A)$, in which case we write $M\models p(\bar{a})$, and <em>omitted by $M$</em> if no $\bar{a}\in M$ realizes $p$;</li>
<li><em>isolated by</em> some satisfiable $L_A$ formula $\psi(\bar{x})$ if $\Th_AM\models\fa\bar{x}(\psi(\bar{x})\rightarrow\phi(\bar{x}))$ for all $\phi\in p$.</li>
</ul>
<p>It will also be convenient to consider the (complete) theory $T\coloneqq\Th_AM$ and talk about $n$-types of $T$ instead of $n$-types of $M$ over $A$; in this case, we write $S_n(T)$ for $S_n^M(A)$.</p>
<h1 id="realizing-and-omitting-types">Realizing and Omitting Types</h1>
<h2 id="realizing-types-in-elementary-extensions">Realizing types in elementary extensions</h2>
<div class="env" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/proposition.svg"><b class="envTitle">Proposition. </b>Every $n$-type $p(\bar{x})$ in $M$ over $A$ can be realized in some elementary extension $N\elemextend M$.</div>

<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect">Proof.</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent">It suffices to show that the $L_M$-theory $\ex\bar{x}p(\bar{x})\cup\Diag_\textrm{el}M$ is satisfiable; by the <a href=https://zhaoshenzhai.github.io/mathwiki/compactness_theorem.md class="internalLink proved_by" title="Compactness Theorem" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/compactness_theorem.md&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:09:38-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:09:38-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:09:38-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Compactness Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/compactness_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-12-15T16:09:38-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:09:38-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:09:38-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Compactness Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/compactness_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/compactness_theorem.md&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:09:38-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:09:38-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:09:38-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Compactness Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/compactness_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Compactness Theorem</a>, it further suffices to show that the $L_M$-sentence $\Delta\coloneqq\ex\bar{x}\phi(\bar{x},\bar{a})\land\psi(\bar{a},\bar{b})$, where $\phi(\bar{x},\bar{a})\in p(\bar{x})$, $M\models\psi(\bar{a},\bar{b})$, $\bar{a}\in A$, and $\bar{b}\in A^c$, is satisfiable.
<br>
  To this end, let $N_0\models\ex\bar{x}p(\bar{x})\cup\Th_AM$. Since $\ex\bar{z}\psi(\bar{a},\bar{z})\in\Th_AM$, we have $N_0\models\ex\bar{x}\phi(\bar{x},\bar{a})\land\ex\bar{z}\psi(\bar{a},\bar{z})$. Interpreting the constants $\bar{b}$ as witnesses to $\ex\bar{z}\psi(\bar{a},\bar{z})$ makes $N_0\models\Delta$, as desired.<span style="float:right;">$\blacksquare$</span></span></div>

<div class="space"></div>
<p><strong>Remark.</strong> In particular, this shows that a type $p(\bar{x})$ is complete iff $p=\tp^N(\bar{a}/A)$ for some elementary extension $N\elemextend M$. The converse follows since $S^M_n(A)=S^N_n(A)$ for any elementary extension $N\elemextend M$. If $p$ is complete, let $N\elemextend M$ realize $p$, i.e. $p\subseteq\tp^N(\bar{a}/A)$ for some $\bar{a}\in N^n$, from which equality follows by completeness of $p$.</p>
<h2 id="the-omitting-types-theorem">The Omitting Types Theorem</h2>
<h1 id="types-of-models">Types of Models</h1>
<h2 id="homogeneous-models">Homogeneous models</h2>
<div class="env" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/lemma.svg"><b class="envTitle">Lemma. </b>If $\bar{a}_1,\bar{a}_2\in M^n$ realize the same types of $M$ over $A$, that is, if $\tp^M(\bar{a}_1/A)=\tp^M(\bar{a}_2/A)$, then there is an elementary extension $N\elemextend M$ and an automorphism $h\in\Aut_AM$ sending $\bar{a}_1\mapsto\bar{a}_2$.</div>

<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect">Proof.</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent"></span></div>

<h2 id="prime-and-atomic-models">Prime and atomic models</h2>
<h2 id="saturated-models">Saturated models</h2>


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                December 12, 2024 | Zhaoshen Zhai

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